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The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological…

We investigate the similarities and differences between the module of symmetric tensors TS^n_A(M) and the module of divided powers \Gamma^n_A(M). There is a canonical map \Gamma^n_A(M) \to TS^n_A(M) which is an isomorphism in many important…

We determine the composition factors of the tensor product $S(E)\otimes S(E)$ of two copies of the symmetric algebra of the natural module $E$ of a general linear group over an algebraically closed field of positive characteristic. Our main…

We study the stable cohomology groups of the mapping class groups of surfaces with twisted coefficients given by the $d^{th}$ exterior powers of the first rational homology of the unit tangent bundles of the surfaces…

We study the vanishing of (co)homology along ring homomorphisms for modules that admit certain filtrations, and generalize a theorem of O. Celikbas-Takahashi. Our work produces new classes of rigid and test modules, in particular over local…

This thesis addresses the theory of topological spaces and the foundations of persistence theory. We will discuss chain complexes and the associated simplicial homology groups, as well as their relationship with singular homology theory.…

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We…

In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered…

We introduce and investigate notions of persistent homology for p-groups and for coclass trees of p-groups. Using computer techniques we show that persistent homology provides fairly strong homological invariants for p-groups of order at…

This paper studies the basic K-theoretic properties of a triangulated persistence category (TPC). This notion was introduced in our earlier papers on triangulation, persistence, and Fukaya categories (arXiv:2304.01785 and arXiv:2104.12258)…

One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…

Multiparameter persistent homology has emerged as a powerful generalization of topological data analysis, capable of encoding multivariate filtrations. However, the algebraic complexity of multiparameter persistence modules, marked by wild…

Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…

Working over the prime field F_p, the structure of the indecomposables Q^* for the action of the algebra of Steenrod reduced powers A(p) on the symmetric power functors S^* is studied by exploiting the theory of strict polynomial functors.…

In this paper, we study further properties and applications of weighted homology and persistent homology. We introduce the Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology. For applications, we show…

In this paper, we introduce a persistent (co)homology theory for Cayley digraph grading. We give the algebraic structures of Cayley-persistence object. Specifically, we consider the module structure of persistent (co)homology and show the…

Exploiting particular features of classical groups, simple constructions are given for the irreducible constituents of the tensor square of the adjoint modules and the leading terms in higher tensor powers. This provides an independent…

We study the cohomology of Aut(F_n) and Out(F_n) with coefficients in the modules \wedge^q H, \wedge H^*, Sym^q H or Sym^q H^*, where H is the Out(F_n)-module obtained by abelianising the free group F_n. For reasons which are not…

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…

We show in this work that homology in degree d of a congruence group, in a very general framework, defines a weakly polynomial functor of degree at most 2d and we describe this functor modulo polynomial functors of smaller degree. Our main…